Network modelling of strong and intermediate wettability on
electrical resistivity and capillary pressure
H.N. Man, X.D. Jing
*Centre for Petroleum Studies, T.H. Huxley School of Environment, Earth Sciences and Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BP, UK
Received 1 December 1999; received in revised form 14 July 2000; accepted 31 August 2000
Abstract
A network model that investigates electrical resistivity and capillary pressure curves of oil/water/rock systems for a full-¯ooded cycle (primary drainage, imbibition and secondary drainage) is presented. This model uses a realistic pore geometry in the form of a grain boundary pore (GBP) shape and pore constrictions. The model also incorporates pore-scale displacement mechanisms and pore-scale wettability alteration that are physically based. A range of contact angles (from 0 to 180°) has been investigated. A detailed description of wettability at the pore scale was simulated to allow both water- and oil-wet regions existing within a single pore. Our numerical simulated results show experimentally observed non-linear trends in double-logarithmic plots of resistivity index vs water saturation. Furthermore, our results show that contact angle hysteresis, which leads to dierent pore scale physics (e.g., snap-o vs piston-like displacement), reveals hysteresis observed in both electrical resistivity and capillary pressure curves. Ó 2001 Elsevier Science Ltd. All rights reserved.
Keywords:Network model; Electrical resistivity; Capillary pressure; Pore geometry; Wettability; Hysteresis
1. Introduction
Initially, all hydrocarbon-bearing reservoirs con-tained rocks that were fully saturated with water. Hydrocarbons may migrate into these regions displacing water and the ¯uids equilibrate over geological periods of time to occupy the pore space. This is provided that the pressure dierential between these two immiscible ¯uids can be overcome. At equilibrium, this pressure dierential known as the capillary pressure,Pc, is related
by the Young±Laplace equation
Pc r
whereris the interfacial tension between the immiscible
¯uids and r1 and r2 represent the principal radii of
curvature normal to each other. In oil/water/rock sys-tems, the capillary pressure is often de®ned as the oil
pressure minus the water pressure (i.e., PcPoÿPw).
The relationship between capillary pressure and water
saturation is important in locating zonal regions, where there is a transition from water to oil of a hydrocarbon reservoir. The balance of capillary against gravitational forces determines initial ¯uid distributions across the transition zone and, together with viscous forces, aects the eciency of oil recovery by water injection.
Capillary pressure controls the distribution of ¯uids. Electrical resistivity of a ¯uid saturated rock depends on the distribution of conducting phase [1±4]. One of the more reliable techniques used to evaluate hydrocarbon potential, in a petroleum reservoir, namely electrical logging is based on an empirical relation called the Archie saturation equation [5]. The equation relates the electrical resistivity of the rock sample to water satura-tion such that
whereRt is the resistivity of the sample at a given water
saturationSw(i.e., partially saturated with water) andRo
is the resistivity of the sample at 100% water saturation.
nis an empirical parameter called the Archie saturation
exponent. To determine n, the gradient of a double
logarithmic plot of resistivity index against water www.elsevier.com/locate/advwatres
*
Corresponding author. Tel.: 7320; fax: +44-20-7594-7444.
E-mail address:[email protected] (X.D. Jing).
saturation is measured. The ratio Rt=Ro is usually
de-noted byI and is called the resistivity index.
Eq. (2) also applies to environmental engineering problems where, for example, electrical measurements may be applied to monitor the level of soil contamina-tion by non-aqueous phases. An understanding of the electrical properties and their relation to ¯uid saturation in soils improves the assessment of contamination and helps to design remedial engineering processes.
Generally, capillary pressure and electrical resistivity curves are a function of saturation history, i.e., which ¯uid is displacing and which ¯uid is displaced, and ex-hibits hysteresis. The degree of hysteresis is found to be dependent on pore structure [2] and wettability [1±4].
Wettability is a term describing which ¯uid amongst at least one other immiscible ¯uid has a tendency to adhere to a surface. It is a very important parameter because the surface properties of the rock determine the ¯uid distribution in the pore space. Ultimately electrical properties, oil displacement and hence oil recovery is aected by the wettability of the rock. A direct tech-nique used by many laboratories to quantify wettability is through the contact angle. In an oil/water/rock sys-tem, it is the angle measured through the water phase by placing a drop of oil or water onto a ¯at solid surface that should be representative of the reservoir considered. A strongly water-wet rock means that water has the greater tendency to spread on the surface. Hence, the contact angle is roughly zero. For a strongly oil-wet rock, the roles of the ¯uids are reversed and the contact
angle is near 180°. Intermediate wettability describes a
rock that has no overall dominant preference to either
¯uid. The contact angle is near 90°.
Hirasaki [6] gave a theoretical insight of wettability from ®rst principals. The stability and thickness of water ®lms determine wetting in oil/water/rock systems. This in turn depends on a parameter known as the disjoining pressure. This additional disjoining pressure is due to the microscopic interactions between atoms that are signi®cant for residual ®lms. The disjoining pressure isotherm has a local maximum: when this is exceeded, the thick water ®lm ruptures to a molecular thin one
[6,7] (Fig. 1). The general consensus is that wettability alteration can take place if oil is left in contact with the rock surface for sucient time. In reality, wettability alteration at the reservoir scale occurs over geological periods of time. The critical capillary pressure, associ-ated with wettability reversal, depends on mineralogy [8], oil and brine properties, such as pH [9], reservoir pressure and temperature [6]. The actual mechanism of wettability reversal is still unclear [10].
Kovscek et al. [7] theorised, in their numerical model, that if regions with a molecular thin water ®lm are contacted by oil, then they are susceptible to adsorption by surface-active components in the oil. Therefore, these regions will be rendered oil-wet (Fig. 1). Kovscek et al. [7] assumed that the contact angle for these oil-wet
surfaces were 180°, while 0 for water-wet surfaces.
However, Kovscek et al.'s [7] theoretical pore-scale model is not a network model, but a model formed by a capillary bundle of tubes, with a cornered cross-section of dierent radii, parallel to each other, but of equal lengths. The radius of each capillary remains constant along its length. The use of pore shape with crevices allows mixed-wettability at the pore scale. This draws on experimental evidence showing that variations of wet-ting can occur within a pore [11].
Recently, numerical simulators have been generalised to allow for any contact angle. It is known that systems rarely exhibit perfectly wetting (water- and oil-wet) conditions [12,13]. Dixit et al. [14] predicted capillary pressure and relative permeability curves incorporating the eects of wettability alteration. They randomly dis-tributed contact angles throughout their network to simulate non-uniform wettability. They found that in a water-wet system, oil recovery increases as the randomly
distributed contact angles approach a limit of 90°. This
seems counter-intuitive, but the pore ®lling sequence, determined by the Young±Laplace equation (Eq. (1)), reveals oil in larger pores is no longer trapped. Exper-imentally Li and Wardlaw [15] have shown, for water-wet systems, snap-o is less favoured as the contact angle increases. This greatly reduces the amount of oil trapped in larger pores. When pores are simulated to
have a contact angle greater than 90°, Dixit et al. [14]
observed oil recovery is dependent on the fraction of pores that are oil-wet.
Blunt [16,17] explicitly modelled the ``co-operative pore ®lling process'', which is a function of contact angle. The process is so-called because the ¯uid dis-placement at the nodal pore where each tube meets is dependent on whether neighbouring tubes meeting at the junction are water- or oil-®lled [18] (Fig. 2). Incor-porating Kovscek et al.'s [7] physical development of mixed-wettability, Blunt [17] was able to reproduce Dixit et al.'s [14] observations even though the contact
angle was the same everywhere. éren et al. [19], too,
modelled the co-operative pore ®lling process and their quantitative predictions of capillary pressure and rela-tive permeability agreed closely with experimental measurements for a reservoir rock.
While numerical simulation of relative permeability and oil recovery has been a topic of active research, there is a lack of theoretical work concerning electrical
resistivity of porous rocks fully or partially saturated by brine. Theoretically, Dicker and Bemelmans [20] realised the importance of the continuity of water to ensure electrical conductance at low water saturations. Wang and Sharma [21], Sharma et al. [22] and Suman and Knight [23] investigated the eects of pore structure and wettability on electrical resistivity.
However, these preceding authors incorporated a circular pore geometry. They were not able to model observed experimental electrical resistivity behaviour, especially at low water saturations for water-wet rocks. Although they implemented the conductive contribution of molecular water ®lms, the curves tended to in®nity below some ®nite water saturation. On the contrary, many authors [2,4,24±27] experimentally observed that the resistivity curves tend to some ®nite resistivity index and curved towards lower values of saturation exponent demonstrating the so-called ``non-Archie'' behaviour.
The eect of a physical development of pore-scale intermediate wettability on capillary pressure and
relative permeability has been studied by Blunt [17,28]
andéren et al. [19]. However, the eect of this detailed
wettability scenario has not been investigated for elec-trical properties. The variation in the saturation expo-nent due to pore structure and wettability has important implications for evaluation of water saturation and, hence, oil in place. Therefore, this paper primarily sim-ulates electrical resistivity and capillary pressure by in-corporating variations of wettability, through a general contact angle, within a single pore. Contact angle hys-teresis may occur due to surface roughness and ad-sorption [13]. In addition, the reservoir pore space is interconnected, pores do not have uniform cross-sec-tions and the pore space can be resolved into two in-terconnected regions: pore throats that interconnect with pore bodies [29].
2. Theoretical model
Our previous three-dimensional network model [30,31] incorporates a physical development of pore-scale wettability on various petrophysical characteristics (in particular electrical resistivity and capillary pressure) starting from primary drainage continuing to spon-taneous/forced imbibition and secondary drainage. In this paper, we have extended the capabilities of the network model to allow contact angles of any value
between 0 and 180°.
The network model features key geometrical attri-butes such as pore connectivity, constrictions along the pore length, pore size distribution and pore shape (Fig. 3). We chose the shape of four closely packed uniform rods, which we refer to as a star-like grain boundary pore (GBP) shape, to represent the cross-section of the conduits (Fig. 3). The constriction, along the length of the conduit, is represented by a sinusoidal variation [21,22,30,31].
In this paper, we de®ne the radius of the nodal pore,
yp, as
ypmax yb1;yb2;. . .;ybk; 3
whereybiis the pore body radius of theith tube attached
to each nodal pore.kis the number of tubes attached to
the node.
3. Network simulation
The numerical solution of the electrical resistivity and the absolute permeability are found by solving sets of simultaneous equations that relate the network ¯ow properties to those of individual pore elements. The procedure starts with initial estimates of voltage and hydraulic potential for each junction. No-¯ow or per-iodic boundary conditions can be imposed in the di-rection perpendicular to the potential gradient. With each pore tube, there is an electrical and hydraulic Fig. 3. Dierent viewpoints (from top to bottom: 3-D, circumferential and transverse) of new pore tube with constrictions. The pore constriction varies according toyybÿ ybÿytsin pLx
ÿ
conductance. The voltages of the junction, where the tubes meet, are calculated using Ohm's and Kircho's laws. The hydraulic potential of the junctions are cal-culated based on laminar ¯ow in pipes and mass con-servation. A ¯ow chart of our algorithm to calculate the electrical resistivity during primary drainage and abso-lute permeability can be found in Fig. 4.
The main approximations that have been made in our network model are:
1. The circumferential radius of curvature of the oil/ water interface near the corner of the crevices is much less than the transverse radius of curvature (Fig. 5). Therefore, the transverse radius of curvature has been
neglected in the calculations using the Young±La-place equation [32].
2. The head meniscus has also been neglected. This is because the radius of the head meniscus is in the order of microns, whereas the length of the pore tube has been calculated to be in the order of hundred mi-crons for a range of realistic porosities of granular sandstones.
3. Although the co-operative pore ®lling process has been modelled, we do not model explicitly the volu-metric and electrical contribution of the pore nodes. These properties are taken into account by the con-stricted pore elements.
4. Displacement cycles
We will now describe the ¯uid displacements occur-ring at the pore scale to calculate the full-¯ooded cycle for electrical resistivity and capillary pressure curves. The primary drainage cycle (in which, for clarity, drainage will be consistently de®ned as ``oil displacing water'' and imbibition as ``water displacing oil'' irre-spective of which ¯uid preferentially wets the solid sur-face) have been discussed mathematically elsewhere [31]. The mathematical analysis describing the pore-scale events that occur during water injection and oil re-in-jection has been extended from strongly wetting con-ditions to any contact angle.
4.1. Primary drainage
At the beginning of each simulation, the network is fully saturated with water. The entire surface of the
network is designated as water-wet with h0. By
in-creasing the capillary pressure, oil displaces water in a piston-like fashion, mimicking oil migration into a res-ervoir. Primary drainage continues until a maximum
capillary pressure is reached. Thick water ®lms that re-side along the water-wet surfaces may rupture during the course of primary drainage to a molecular thin ®lm. The contact angle is no longer zero here [6,7]. We may account for contact angle hysteresis by assuming that all regions with a molecular thin ®lm after primary drain-age are assigned a new contact angle.
To calculate capillary pressure and electrical re-sistivity curves, the pore tube is divided into sections of in®nitesimal constant thickness by making cuts
per-pendicular to the x-axis. An increase in the capillary
pressure pushes the oil/water interface in discrete steps to the opposite end of the pore tube. Cross-sectionally, the oil/water interface moves towards the crevices (Fig. 5).
Under conditions of capillary equilibrium, the ca-pillary pressure is constant across the oil/water interface. This in turn determines the meniscus shape. The amount of water retained in the crevices can be expressed analytically [33], but because our pore tubes are con-stricted, the volume of water and electrical resistance of each individual tube is found by integrating along the length of the pore tube. The mathematical details can be found elsewhere [31]. Capillary equilibrium is acceptable as long as the capillary number is not greater than about
10ÿ6±10ÿ7 [16,17]. Most reservoir displacements have
capillary numbers less than this. Oil may only enter a tube if one of the adjacent tubes has already been penetrated by oil or it is next to the face where oil is injected. A ¯ow chart of our algorithm to derive both the electrical resistivity and capillary pressure curves during primary drainage can be found in Fig. 4.
The drainage process in the simulations stops at a desired maximum capillary pressure. To investigate the eect of saturation history, water was imbibed into the partially oil-saturated model.
4.2. Water imbibition
4.2.1. Pore-scale mechanisms
Water may advance into an oil-®lled tube in a piston-like fashion dictated by Eq. (4)
PcK1 h r
y; 4
whereK1 his a constant for a particular contact angle
and pore geometry andyis the grain radius. Table 1 lists
the values of K1 h for a GBP shape for a range of
contact angles [33].
Water may also displace oil by snap-o. As the ca-pillary pressure is decreased, water will spontaneously imbibe away from the crevices. Further increases in wetting ¯uid may result in non-wetting ¯uid losing contact with the pore wall [34] (Fig. 6). The non-wetting ¯uid then becomes disconnected (snapped-o) retreating to leave much of the adjoining pores ®lled with non-Fig. 5. Schematic picture showing oil moving into a GBP shape
wetting ¯uid. For a strong water-wet GBP shape
geometry (i.e., h0), snap-o occurs at a capillary
pressure
For an advancing (water displacing oil) contact angle greater than zero, the oil/water meniscus at the crevices is pinned at the position it attained at the end of primary drainage. The pinned oil/water meniscus may start to
move towards the centre of the pore space oncehhow
wherehow is the contact angle of regions after primary
drainage that were in contact with oil. The capillary pressure at which the pinned oil/water meniscus starts to move is stated below and derived in Appendix A.
Ifhow6w90°, then
wherewis the angle that describes the movement of the
oil/water meniscus, ww at PcPcmax, the maximum
imposed capillary pressure during primary drainage
(Fig. 7), and hpd is the contact angle of regions with a
molecular thin ®lm during primary drainage.
Provided how<w, the oil/water meniscus at the
crevices may start to move towards the centre of the pore space at a positive capillary pressure and may meet the other oil/water meniscus at a positive capillary pressure. Subsequently, the oil snaps o at a capillary pressure [31]
o will occur before piston-like advance if
K2 h
We note that there is a constraint on the constriction factor yb=yt
Previously, we have shown that a constriction factor of 2 agrees very well with experimental data [30]. Hence, in this case, snap-o always occurs before piston-like advance for strongly water-wet systems. However, if we
increase the contact angle slightly to h20°, Eq. (9)
becomes
yb
yt
>3:01: 11
Therefore, if we use a constriction factor of 2, then using
h20°, piston-like advance has precedence over
snap-o. Snap-o can still occur, though, if piston-like advance cannot. This situation arises when water cannot piston-like advance in the tube being considered for displacement, because it is not adjacent to a water-®lled node.
Fig. 6. (a) In water imbibition, as water saturation increases, oil may form an inscribed circle within a water-wet pore leading to snap-o; (b) a further increase in water pressure would result in oil retreating towards the pore body. How far the oil ganglia retreat is given by the equation shown.
Table 1
Piston-like advance factor for particular contact angles
Contact angle,h° Piston-like advance factor,K1 ha
Ifhow>w, then the pinned oil/water meniscus moves
at a negative capillary pressure. Now, for this case, water
invasion is forced and oncehhow, the pore space
im-mediately ®lls with water due to the instability of the oil/ water meniscus [17,19]. However, this mechanism is less favourable than water piston-like displacing oil.
Now, water may piston-like advance into a node and has been shown to depend on the number of neigh-bouring tubes ®lled with oil [18] (Fig. 2). For this reason,
the mechanism is a co-operative process. Blunt [28] modelled this invasion, where he used a square cross-section to represent the conduits, using a parametric model given by
Pc
2rcosh
r ÿr
Xn
i1
Aixi; 12
where r is the largest inscribed circle within the pore
shape, n the number of neighbouring tubes ®lled with
(a)
(b)
oil,xi the random numbers between 0 and 1 andAi are
matching parameters. Experiments show signi®cant forced water invasion can occur for samples that exhibit
a contact angle of approximately 60° or greater [35].
Therefore, Blunt [28] chose A1A2 A5
0:015 lmÿ1 to reproduce this eect.
For our nodal pores, Eq. (12) is slightly dierent
Pc K1 h
as Dullien's [35] observation. A node can be water-®lled only if (1) the imposed capillary pressure is lower than the capillary pressure given by Eq. (13), (2) it is adjacent to a water-®lled tube and (3) oil can escape to the outlet. Water may also displace oil in the nodal pores
by snap-o. In this case, Eq. (7) is again used with y
substituted foryp.
4.2.2. Oil layer breakage
During the forced water imbibition of water dis-placing oil, ``oil layers'' may form between the water in the crevices and the water in the pore centre (Fig. 8).
According toéren et al. [19], this occurs when
hP180°ÿw: 14
If the oil/water meniscus on either side of the oil layer touch, then the oil layer collapses preventing the ¯ow of oil through the tube (Fig. 9). However, we assume that a molecular thin oil ®lm still endures after the menisci touch [7,17]. For a general contact anglehow, this occurs
when the distancesBC for the inner and outer oil/water
menisci are identical, i.e., BC inner BC outer
(Fig. 7). The capillary pressure at which the oil layer collapses has been, again, deferred to Appendix A.
4.2.3. Cross-sectional area of water with oil layer If oil layers are present within an in®nitesimal thin section, then the calculation of the cross-sectional area
of water is very similar to the calculation during primary drainage, but with the roles of oil and water reversed. Eectively, the cross-sectional area of water of a section with an oil layer has two components. The water im-bibing away from the crevices, i.e., the outer oil/water meniscus that is pinned at the wettability discontinuity, and the inner oil/water meniscus that is not pinned (Fig. 8). The calculation of the cross-sectional area of water with an oil layer has been simpli®ed, for any contact angle, and is shown in Appendix A.
4.3. Secondary drainage
During oil re-injection (secondary drainage), the capillary pressure is increased from a high negative value to a high positive value. Accordingly, pores that exhibit oil-wet regions spontaneously imbibe oil ®rst.
Competition between the snap-o and piston-like displacement mechanism also occurs during oil re-injection. This is very much analogous to water injection with the roles of oil and water reversed and, for brevity, we will not repeat the mathematical derivations. How-ever, the snap-o mechanism may only occur if the in-®nitesimally thin section contains an oil layer or the molecular thin oil ®lm that persisted after oil layer breakage during forced imbibition. In tubes with a molecular thin oil ®lm, the oil layer will spontaneously re-form at the same capillary pressure at which it broke (given by Eq. (A.13) in Appendix A). As the capillary pressure is increased, the oil layer grows larger. The outer meniscus of the oil layer moves towards the crevices and the inner meniscus moves towards the centre of the pore space. Cross-sectionally, the displacement process is the reversal of Fig. 9.
The oil layer could swell large enough such that the oil/water menisci may meet one another analogous to oil becoming unstable during spontaneous imbibition. Here, then, water snaps-o at a capillary pressure
Pc
When this happens, the centre of the thin section ®lls with oil (Fig. 10). Displacement is only possible, if there is continuity of oil from the inlet to the tube being considered for displacement either via tubes that are penetrated by oil or oil layer spanning tubes. Further-more, if the tube contains an oil layer, then it must be adjacent to a totally water-®lled section for the central portion of water to escape. If the tube being considered for displacement is not adjacent to a totally water-®lled section, water cannot escape and remains trapped. Since our chosen pore shape contains crevices, water can only remain trapped in the centre of the pore space. Ac-cordingly, the contribution to the electrical conductance is zero. Here, then, only water residing in the crevices that always remain connected contributes to the elec-trical conductance.
In terms of pore ®lling, neighbouring tubes again in¯uence the capillary pressure for ®lling the pore with oil. But this time, water inhibits the procedure [28]
PcK1 h r
yp
rX
n
i1
Aixi; 16
wherenis the number of neighbouring tubes ®lled with
water.
Snap-o of water by oil, in the nodal pores, is given by Eq. (15) withy substituted for yp.
5. Results and discussion
We will now present simulations investigating the eects of contact angle on electrical resistivity and Fig. 10. Water may form an inscribed circle within a mixed-wet pore leading to snap-o. Water in the centre of the pore space is, then, trapped but the water in the crevices is always connected.
Table 2
Input parameters used in the network model for Figs. 11±16a
Figure 11 12 13 14 15 16
Contact angle after primary drainage (°) 0 40 80 100 140 180 a
Porosity20%. Modi®ed Rayleigh pore size distribution with minimum, maximum radii and skewness factor0, 100 and 1250lm, respectively.
Constriction factor2. Co-ordination number6.
capillary pressure curves. Table 2 lists the input pa-rameters used in our simulations. We represented the pore structure of our network with a modi®ed Rayleigh distribution. We used the same values of minimum/ maximum radii and skewness factor as those adopted by McDougall and Sorbie [36]. The same authors argued that this distribution gave more realistic pore size dis-tributions of sandstone rocks. In the simulations here,
Pmax
c was chosen such that a portion of all pore tubes are
not penetrated by oil in order to achieve initial water saturation.
Table 3 presents the simulated formation factor, cementation exponent, absolute permeability and satu-ration exponent for this particular rock type. Less-cemented sands usually have high porosities, and from Archie [5], low formation factors: the porosity raised to the power of the cementation exponent equals the re-ciprocal of the formation factor. The cementation exponent, amongst other factors, is a function of shape, sorting, packing of the grains, overburden pressure, presence of clay and reservoir temperature. The simu-lated formation factor and cementation exponent pre-sented here are typical of those measured experimentally
by Moss and Jing [4] of sandstone samples. The satu-ration exponent during primary drainage was calculated by considering the line of best ®t between the saturation range 30±100%.
We ®rst simulated a case that exhibits no contact angle hysteresis following the primary drainage cycle,
i.e., h0 throughout the simulation (Fig. 11). During
spontaneous imbibition, the resistivity indices are slightly lower than primary drainage (Fig. 11). This is because during primary drainage, oil migrates into the system as a front. In spontaneous imbibition, oil is already in place, but because snap-o is increasingly favourable as the contact angle decreases, the oil tends to exist as isolated ganglia. Therefore, there are more large open spaces for the electrical current to ¯ow through the sample. This type of hysteresis has been observed experimentally [1,37].
Note that the saturation exponent is at the lower range of saturation exponents observed experimentally. Representing all the conduits of the pore space with a cross-sectional GBP shape will retain appreciable amounts of capillary-bound water due to the sharp crevices. In fact, scanning electron microscope (SEM) studies on quartzose sandstones show that overgrowths can produce relatively smooth surfaces with no sharp corners. We matched the saturation exponent, and other electrical and ¯uid ¯ow experimental data for a North Sea reservoir core sample, by assigning GBP and some pores with a circular cross-sectional area [31]. This re-duces the amount of capillary-bound water in the sys-tem. Detailed petrographic analysis on speci®c rock
Fig. 11. Capillary pressure and electrical resistivity curves whenhow0.
Table 3
Simulated petrophysical output data for this particular rock type
Figures 11±16
types will provide information on the degree of cemen-tation and alteration of sharp crevices.
Realistically though, experiments show that advanc-ing and recedadvanc-ing contact angles are generally not identical [12]. We introduce contact angle hysteresis by increasing the contact angle to dierent values for a
number of simulations. We started by inputtingh40°
following the primary drainage cycle.
From the capillary pressure curve (Fig. 12), we notice that during imbibition more water imbibes into the system compared to the strongly water-wet case (Fig. 11). This is because snap-o is almost suppressed for this contact angle (snap-o is totally suppressed
when h45°) allowing the piston-like advance
dis-placement mechanism to dominate at the pore scale. Morrow et al. [38] experimentally observed that weakly water-wet cores gave improved oil recoveries compared to strongly water-wet cores. As commented in the pre-ceding paragraph, snap-o leaves oil existing as isolated ganglia and, therefore, remains trapped during sponta-neous imbibition. Conversely, the suppression of snap-o allows the water to migrate as a front that in turn displaces the well-connected oil.
In the resistivity curve, the cycles following primary drainage shifts upward (Fig. 12). Note that for the strongly water-wet case, the resistivity indices during secondary drainage are lower than that during primary drainage. In Fig. 12, there does not appear to be much hysteresis between the three cycles compared to the strongly water-wet case. The increase in the resistivity indices, again, indicates oil being well connected and not
existing as isolated ganglia that allowed water to con-duct more eectively for the strongly water-wet case.
In Fig. 13, we simulated a weakly water-wet system
with a contact angle ofh80°. Here, snap-o does not
occur at all. We note, in the resistivity plot, the water imbibition cycle is even higher than the primary drain-age cycle compared to Fig. 12. The trend appears be-cause of the dominance of water piston-like displacing oil. However, we now see a reversal of trend during secondary drainage: the resistivity indices during this cycle are now lower than the water imbibition cycle. This could be explained by the fact that the suppression of snap-o allows piston-like displacement to leave the system almost water-®lled at the end of spontaneous imbibition. Therefore, re-injecting with oil is very similar to the displacement process occurring during primary drainage. Note that there is some saturation change during negative capillary pressures even though the
contact angle is less than 90°. This is a consequence of
the co-operative pore ®lling process and agrees with Dullien's [35] experimental observation. Water may only enter some nodes at negative capillary pressures (cf. Eq. (13)).
Fig. 14 presents a weakly oil-wet system with a
con-tact angleh100°. During water imbibition, signi®cant
saturation changes occur only for negative capillary pressures. For oil re-injection, there is some saturation change during positive capillary pressures. Again, analogous to the weakly water-wet case, this is due to
the co-operative pore ®lling process. Comparing
previous resistivity curves, we note that now at the
beginning of water imbibition, the slope is much less. This is because water is mainly entering large pores, but not penetrating them. Once the tubes are penetrated by water, we then see a signi®cant drop in the resistivity indices. For water-wet samples, we do not observe this two-tier gradient regime. During spontaneous
imbi-bition, once water enters the pore body of a constricted tube, water spontaneously penetrates the whole tube.
Furthermore, the hysteresis between the water im-bibition and secondary drainage resistivity indices ap-pear to be less than for a weakly water-wet system (Fig. 13). For a weakly oil-wet system, water invades the Fig. 13. Capillary pressure and electrical resistivity curves whenhow80°.
largest pores ®rst and advances dendritically [17]. Note that oil layers do not form at all (cf. Eq. (14)). This leads to more trapping of oil in the smallest pore tubes. Oil being trapped in the smallest pore tubes lead to higher resistivity indices during secondary drainage and a higher residual oil saturation compared to the weakly water-wet case.
In Fig. 15, we increased the contact angle even further
to h140°. Oil layers can form now. Oil layers can
provide continuity for the oil to escape. Observe that the residual oil saturation at the end of water injection is lower than for the weakly oil-wet system (Fig. 14). Since oil layers may be present, water may snap-o. Note that the resistivity indices during secondary drainage are Fig. 15. Capillary pressure and electrical resistivity curves whenhow140°.
visibly lower. The hysteresis between the resistivity in-dices during water imbibition and secondary drainage is greater compared to the weakly oil-wet case. This is due to the formation of oil layers that allow less trapping of oil in smaller pore tubes.
For a strongly oil-wet case (Fig. 16), h180°, the
residual oil saturation is even less. This is because a very large negative capillary pressure is needed in order to break the oil layers. Analogous to a strongly water-wet
sample, i.e.,h0, snap-o has precedence over
piston-like advance during secondary drainage. This implies oil does not invade the system in the form of piston-like advance, but spontaneously displaces water throughout the sample. Snap-o occurs, analogous to spontaneous imbibition, for the smallest pore tubes ®rst. The dra-matic increase in the resistivity indices during secondary drainage is a result of the oil occupying the smallest pore tubes (Fig. 16). In fact, there appears to be virtually no hysteresis between the resistivity indices during water imbibition and secondary drainage (Fig. 16).
6. Conclusions
We have presented a network model based on a physical development of pore-scale wettability with the main intention to simulate electrical resistivity and capillary pressure of oil/water/rock systems, in a full drainage and imbibition cycle, for any contact angle from 0°to 180°.
For a strongly water-wet system, snap-o is the dominant event occurring at the pore scale, and oil tends to exist as isolated ganglia during spontaneous imbi-bition. Therefore, strongly water-wet samples are more conductive during spontaneous imbibition than primary drainage.
During water imbibition, for a weakly water-wet system, the resistivity indices are high, because as the contact angle increases, snap-o is suppressed. Oil no longer exists as isolated ganglia that allow the sample as conductive as during primary drainage. The suppression of snap-o also results in the weakly water-wet system to be almost water-®lled at the end of water imbibition. Therefore, re-injecting with oil almost mimics the pri-mary drainage process. The resistivity indices during secondary drainage maybe even as low as those attained during primary drainage.
Increasing the contact angle further into the weakly oil-wet regime, during water imbibition, water displaces oil from the largest pores ®rst. Oil layer formation is not possible and leads to trapping of oil in the smallest pores. The resistivity indices during secondary drainage can be quite high and as high as those attained during water imbibition.
When formation of oil layers is possible (strongly oil-wet), oil in smaller pores may escape to the outlet
through these oil layers. Oil entrapment, in this case, is less than for a weakly oil-wet system. This explains why the hysteresis between the resistivity indices during water imbibition and secondary drainage is actually greater than for a weakly oil-wet case. When snap-o of
water dominates (i.e.,h180°) there, again, appears to
be no hysteresis between water imbibition and second-ary drainage similar to the weakly oil-wet case. How-ever, the saturation exponents during water imbibition and secondary drainage are much higher than that during primary drainage. This is in close agreement with experimental observations. Our results for a variety of contact angles indicate that the saturation exponent during secondary drainage does not monotonically in-crease fromh0°toh180°.
Acknowledgements
Hing Man would like to thank Martin Blunt for discussion on the co-operative pore ®lling process. We thank EPSRC for ®nancial support.
Appendix A
A.1. Pore-scale mechanisms
The pinned oil/water meniscus may move during
water imbibition (Fig. 7) oncehhow. That is,
We note that this is in agreement with éren et al. [19].
They used triangles to represent the cross-section of
their conduits (Fig. 17). They labelled bl as the largest
half-angle of their triangle. Accordingly, our w is
re-lated tobl via (Fig. 17)
bl90°ÿw: A:3
Substituting this bl into their derived expression for
when the pinned oil/water meniscus moves [19], yields our result.
the oil/water contact moves, the imposed capillary pressure, Pimp
c , is low enough for the oil/water contact
to move further towards the throat of the tube. The furthest position the oil/water contact moves is denoted
byxxmove. For non-zero values ofhpd; w cannot be
explicitly solved in terms of Pc. Therefore, a function
f Pc;w 0 was de®ned in order to solve forwvia the
Newton±Raphson bisection method [39]. Hence, if ÿ90°6wÿhow60, then
hence, the furthest position of the oil/water interface moving towards the pore throat is,
xmove
A.2. Oil layer breakage
The capillary pressure at which the oil layer collapses occurs whenBC inner BC outer(Fig. 7).
andPc<0, because water is being forced into the system.
We now solve for the capillary pressure,Pc, at which
the oil/water menisci touch by equating Eq. (A.8) with Eq. (A.9). However, we cannot express Eq. (A.8) in
terms of w or Eq. (A.9) in terms of w. But we may
eliminate Pc to obtain,
BC inner r
Fig. 17. (a) éren et al. [19] used triangles to model their pore shape whereblis the largest half-angle of the triangle; (b) we can relatew
r
oil/water menisci will touch at yyb ®rst. From
Eq. (A.10c),wis known and so we can solve for wby
de®ning
f w;w BC inner ÿBC outer 0: A:14
Again, the hybrid root-solver known as the ``Newton± Raphson bisection method'' [39] was used to determine
w,
Hence,Pc is calculated by using Eq. (A.13).
However, in general, once the oil/water menisci touch at yyb, the imposed capillary pressure, Pcimp, is low
enough for the oil lens to break further towards the throat of the pore tube (Fig. 18). Eq. (A.14) can be used again to determine the furthest position of oil lens
breakage denoted by xxbreak in Fig. 18. An initial
guess wnew is fed into Eq. (A.14) and once again w is
determined by the Newton±Raphson bisection method. Then, a new capillary pressure under equilibrium con-ditions,Pnew
c , is calculated via Eq. (A.13) such that
Pcnew ÿ
whereeis the tolerance then we acceptwnewandxbreakis
given by
A.3. Cross-sectional area of water with oil layer
Fig. 9 illustrates the cross-sectional area of water with an oil layer. The outer oil/water meniscus that is pinned at the wettability discontinuity contributes an area of water of (Fig. 9)
The inner oil/water meniscus, which is not pinned, contributes a cross-sectional area of water of (Fig. 9)
Awater2y2 4ÿp ÿ4y2
AwaterAwater1Awater2. Note that we cannot explicitly
express Awater in terms of one independent variable
un-lesshow180°. Equating Eq. (A.10a) with Eq. (A.10c),
we may numerically solve
ÿsin1 whow
for w and substitute into Awater2. However, to simplify
our calculations, we henceforth approximate that the cross-sectional area of water is dominated by water Fig. 18. The oil/water menisci on either side of the oil layer ®rst meet
entering the centre of the pore space, i.e., we neglect
Awater1. Hence, the volume of water in the constricted
pore tube is The electrical resistance of the constricted pore tube can be found in a similar fashion
R
where we have given that the resistivity of water is unity, as it cancels out in the resistivity index calculations.
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